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Let $p$ be a prime number of the form $4k+1$. Fermat's theorem asserts that $p$ is a sum of two squares, $p=x^2+y^2$.

There are different proofs of this statement (descent, Gaussian integers,...). And recently I've learned there is the following explicit formula (due to Gauss): $x=\frac12\binom{2k}k\pmod p$, $y=(2k)!x\pmod p$ ($|x|,|y|<p/2$). But how to prove it?

Remark. In another thread Matt E also mentions a formula $$ x=\frac12\sum\limits_{t\in\mathbb F_p}\left(\frac{t^3-t}{p}\right). $$ Since $\left(\dfrac{t^3-t}p\right)=\left(\dfrac{t-t^{-1}}p\right)=(t-t^{-1})^{2k}\mod p$ (and $\sum t^i=0$ when $0<i<p-1$), this is, actually, the same formula (up to a sign).

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Keep in mind that binomial coefficients are always integers -- think of Pascal's triangle. Also, the "middle" of Pascal's triangle is always an even number so $1/2$ of $2k$ choose $k$ is always an integer. –  Bill Cook Oct 19 '11 at 22:01
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What do you mean by "what to make of"? Do you mean, why is it true? Are you asking for a proof of the formula? –  Gerry Myerson Oct 19 '11 at 22:07
    
@GerryMyerson Well, yes, asking "how to prove it" I'm asking for a proof. –  Grigory M Oct 19 '11 at 22:16
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By the way, once you have an explicit value for $i \in \mathbb{F}_p$, then simply $x^2 + y^2 = (x + iy)(x - iy)$. –  JavaMan Oct 20 '11 at 1:15
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4 Answers

up vote 10 down vote accepted

Here is a high level proof. I assume it can be done in a more elementary way. Chapter 3 of Silverman's Arithmetic of Elliptic Curves is a good reference for the ideas I am using.

Let $E$ be the elliptic curve $y^2 = x^3+x$. By a theorem of Weyl, the number of points on $E$ over $\mathbb{F}_p$ is $p- \alpha- \overline{\alpha}$ where $\alpha$ is an algebraic integer satisfying $\alpha \overline{\alpha} =p$, and the bar is complex conjugation. (If you count the point at $\infty$, then the formula should be $p - \alpha - \overline{\alpha} +1$.)

Let $p \equiv 1 \mod 4$. We will establish two key claims: Claim 1: $\alpha$ is of the form $a+bi$, for integers $a$ and $b$, and Claim 2: $-2a \equiv \binom{(p-1)/2}{(p-1)/4} \mod p$. So $a^2+b^2 = p$ and $a \equiv -\frac{1}{2} \binom{(p-1)/2}{(p-1)/4}$, as desired.

Proof sketch of Claim 1: Let $R$ be the endomorphism ring of $E$ over $\mathbb{F}_p$. Let $j$ be a square root of $-1$ in $\mathbb{F}_p$. Two of the elements of $R$ are $F: (x,y) \mapsto (x^p, y^p)$ and $J: (x,y) \mapsto (-x,jy)$.

Note that $F$ and $J$ commute; this uses that $j^p = j$, which is true because $p \equiv 1 \mod 4$. So $F$ and $J$ generate a commutative subring of $R$. If you look at the list of possible endomorphism rings of elliptic curves, you'll see that such a subring must be of rank $\leq 2$, and $J$ already generates a subring of rank $2$. (See section 3.3 in Silverman.) So $F$ is integral over the subring generated by $J$. That ring is $\mathbb{Z}[J]/\langle J^2=-1 \rangle$, which is integrally closed. So $F$ is in that ring, meaning $F = a+bJ$ for some integers $a$ and $b$.

If you understand the connection between Frobenius actions and points of $E$ over $\mathbb{F}_p$, this shows that $\alpha = a+bi$.

Proof sketch of Claim 2: The number of points on $E$ over $\mathbb{F}_p$ is congruent modulo $p$ to the coefficient of $x^{p-1}$ in $(x^3+x)^{(p-1)/2}$ (section 3.4 in Silverman). This coefficient is $\binom{(p-1)/2}{(p-1)/4}$. So $$- \alpha - \overline{\alpha} \equiv \binom{(p-1)/2}{(p-1)/4} \mod p$$ or $$-2a \equiv \binom{(p-1)/2}{(p-1)/4} \mod p$$ as desired.

Remark: This is very related to the formula Matt E mentions. For $u \in \mathbb{F}_p$, the number of square roots of $u$ in $\mathbb{F}_p$ is $1+\left( \frac{u}{p} \right)$. So the number of points on $E$ is $$p+\sum_{x \in \mathbb{F}_p} \left( \frac{x^3+x}{p} \right).$$

This is essentially Matt's sum; if you want, you could use the elliptic curve $y^2 = x^3-x$ in order to make things exactly match, although that would introduce some signs in other places. So your remark gives another (morally, the same) proof of Claim 2.

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Shouldn’t this ‘Weyl’ be ‘Weil’? –  Grigory M Jan 2 at 21:54
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(Below is a kind of a low-tech version of David Speyer's answer — [essentially] proving Weil for a cubic instead of using it. On the other hand, it's a kind of a more conceptual version of the computation with Jacobsthal sums.)


I. Preliminaries

For any two (nontrivial multiplicative) characters mod $p$ define Jacobi sum $$ J(\chi,\lambda):=\sum_{a+b=1}\chi(a)\lambda(b). $$ Recall that (if $\chi\lambda$ is a non-trivial character) $$ J(\chi,\lambda)=\frac{g(\chi)g(\lambda)}{g(\chi\lambda)}, $$ where $$ g(\chi)=\sum\chi(t)\zeta^t $$ is the usual Gauss sum (cf. beta-function and gamma-function, btw).

Corollary. Since $|g(\chi)|=\sqrt p$, the absolute value of a Jacobi sum is $\sqrt p$ (given that $\chi$, $\lambda$ and $\chi\lambda$ are non-trivial).

II. Proof

Take $\chi$ to be a non-trivial character of order $4$ mod $p=4k+1$; $\chi^2=:\rho$ is then the quadratic character. Since $\chi$ and $\rho$ take values in $\{\pm1,\pm i\}$, the sum $J(\chi,\rho)$ is a Gaussian integer, and by Corollary it has norm $p$.

This already gives a proof of Fermat's $4k+1$ theorem. And explicitly (using that $J$ is real only if $a$ is a quadratic residue) $$ \operatorname{Re}J(\chi,\rho)=\frac12\sum_t\left(\frac tp\right)\left(\frac{1-t^2}p\right)=\frac12\sum_t\left(\frac{t-t^3}p\right). $$ Which is, of course, just an example of counting of points on an elliptic curve ($y^2=x^3-x$ or $y^2=1-x^4$) — well, counting points on hypersurfaces is the point of Jacobi sums, after all.


Reference: Ireland, Rosen. A Classical Introduction to Modern Number Theory (ch. 8).

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Denote by $\phi(D)$ the Jacobsthal sum $$ \sum_D\left(\frac tp\right)\left(\frac{t^2+D}p\right). $$ Note that $\phi(a^2D)=(a/p)\phi(D)$, so $|\phi(D)|$ depends only on $\bigl(\frac Dp\bigr)$; let $2x$ and $2y$ be the absolute values of the Jacobsthal sum for quadratic residue and quadratic non-residue respectively.

Theorem (Jacobsthal, 1907). If $p=4k+1$, $x^2+y^2=p$.

(As explained in the OP, this statement implies the formula of Gauss.)


Proof. Obviously, $$ \sum_{D\in\mathbb F_p}\phi(D)^2=2(p-1)(x^2+y^2), $$ so we need to compute the sum $$ \sum_{D}\phi(D)^2= \sum_{D,s,t}\left(\frac{st}p\right)\left(\frac{s^2+D}p\right)\left(\frac{t^2+D}p\right)= \sum_{s,t}\left(\frac{st}p\right)\sum_D\left(\frac{s^2+D}p\right)\left(\frac{t^2+D}p\right). $$

Lemma. $$ \sum_D\left(\frac Dp\right)\left(\frac{D+c}p\right)=\begin{cases} \phantom p-1,&c\neq 0;\\ p-1,&c=0. \end{cases} $$

To prove the lemma for $c\neq0$ observe that in this case $$ \sum_D\left(\frac{D^2+cD}p\right)= \sum_{D\neq0}\left(\frac{1+cD^{-1}}p\right)= \sum_{D'\neq1}\left(\frac{D'}p\right)=-\left(\frac1p\right)=-1. $$

Now let us apply the lemma to our sum: $$ \sum_D\left(\frac{s^2+D}p\right)\left(\frac{t^2+D}p\right)= \sum_{D'}\left(\frac{D'}p\right)\left(\frac{D'+t^2-s^2}p\right)= \begin{cases} \phantom p-1,&s\neq\pm t;\\ p-1,&s=\pm t; \end{cases} $$ so $$ \sum_{D}\phi(D)^2=(p-1)\left\{\sum_{s=t}\left(\frac{t^2}p\right)+\sum_{s=-t}\left(\frac{-t^2}p\right)\right\}+(-1)\sum_{s\neq\pm t}\left(\frac{st}p\right). $$ For $p=4k+1$ this equals $2p(p-1)$ (the last sum being zero). Hence indeed $x^2+y^2=p$.


(To my surprise, although the proof is quite short and elementary, I haven't been able to find a concise modern reference. Anyway, Jacobsthal's paper «Über die Darstellung der Primzahlen der Form $4n+1$ als Summe zweier Quadrate» is quite accessible.)

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There is a proof on page 192 of Franz Lemmermeyer's book, Reciprocity Laws from Euler to Eisenstein. I found it by typing "sum of two squares" and "binomial coefficient" into Google.

There is also a proof in Allan Adler's paper, Eisenstein and the Jacobian varieties of Fermat curves, which paper is freely available on the web.

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Thank you, I'll try to read it (but if you could post some kind of plan of the proof, it would be very nice). –  Grigory M Oct 20 '11 at 10:14
    
Neither proof is all that easy to summarize. Gauss' original proof, using cyclotomy, was 28 pages long. Adler follows Gauss, trying to streamline it, but it still takes several pages. Lemmermeyer's proof is short but it depends on results previously proved about Jacobi sums. –  Gerry Myerson Oct 20 '11 at 11:56
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